Solving for X: Understanding the Algebraic Expression 4x × 5x 40

In this article, we delve into the algebraic expression 4x × 5x 40 and explore the steps to find the value of x. We'll look at various methods to solve for x, from square roots to simplifying expressions, ensuring a comprehensive understanding of the algebra involved.

Introduction

The expression 4x × 5x 40 is a typical algebraic problem that tests one's ability to manipulate and solve equations. It involves the multiplication of variables and constants, resulting in a quadratic equation. Let's break down the problem and its solutions.

Method 1: Standard Algebraic Manipulation

The most straightforward approach is to simplify the given equation using basic algebraic operations. Let's start with the given equation:

4x × 5x 40

First, simplify the multiplication on the left-hand side (LHS):

22 40

Next, we isolate x2 by dividing both sides by 20:

x2 2

Finally, solve for x by taking the square root of both sides:

x ±√2

Alternative Methods

Method 2: Simplification of Bases

The alternative method involves the bases and exponents approach. Let's rewrite the given expression:

4x × 5x 40

Recognize that 4 and 5 can be expressed in terms of 2:

22 × 22 × 5x 40

Combine the exponents:

24x2 40

Divide both sides by 20:

22x2 40

This simplifies to:

x2 2

Take the square root of both sides:

x ±√2

Method 3: Direct Multiplication

Another method involves directly multiplying and then simplifying:

4x × 5x 40

First, multiply the numbers and the variable:

22 40

Divide both sides by 20:

x2 2

Take the square root of both sides:

x ±√2

Verification and Proof

To confirm the solution, substitute x √2 and x -√2 back into the original equation:

4x × 5x 40

4(√2) × 5(√2) 20 × 2 40

4(-√2) × 5(-√2) 20 × 2 40

The solutions are consistent with the original equation, confirming the accuracy of the solution.

Conclusion

The value of x in the equation 4x × 5x 40 is ±√2. By exploring different methods, we can see that the algebraic manipulation, simplifying bases, and direct multiplication all yield the same result.

References

MathIsFun - Solving Quadratic Equations Khan Academy - Solving Quadratic Equations

Videos

Further Reading

For further reading on similar algebraic expressions and solving for x, please refer to the following articles:

Math Warehouse - One-Step Equations Britannica - Algebra